VLSI ISSCC. Nature, Science. Fig. 1. (entrainment) , (PD) 1996

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Ζηνοβία Παναγιώτα Βασιλόπουλος
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71 1 1 (injection locking) ().,,,,., Fig.

1 (injection locking) ().,,,,., Fig. 1 ( ) PC VLSI ISSCC,, ( [1]) Nature, Science ( ) (entrainment) ( ), (PD) (, ) ( ), 3, 5 ( )

2 7 [][3] (A) (B) () (C) Fig. 1 : (locking range) ( ) Fig. A ω ( ) 1 ( ) [4] [5] [6] 1 Fig. (A) (B) (C) dα = ω ϵ sin α (1) α ω ( [7][8]) dx = F (x) + I(t) () x( R n ) I(t)( R n ) ( ) I(t)

3 73 VOL. 0 NO. JUN f(t) R x 0 ( R n ) ψ( [0, π] S) ( ) f () x = x 0 ψ dψ = ω + Z(ψ) f(t) (3) ω Z(ψ) Ω( ω) ψ ϕ = ψ Ωt ϕ dϕ = ω + Γ(ϕ), (4) Γ(ϕ) = Z(θ + ϕ)f(θ) θ 1 1 Γ(ϕ) ψ (4) [9] [3] (1) (4) Fig. B ϕ dϕ/ = 0 Γ (ϕ) < 0 Γ(ϕ + ) Γ(ϕ ) ( ω) 1:1 ( ω:ω 1:1) m:n ( ω:ω m:n) dϕ = ω+γ m/n(ϕ), (5) Γ m/n (ϕ)= Z(mθ+ϕ)f(nθ) Z 1 1 π π dθ π ISF[10] PPV[11]. Z [1][13] [14] (SPICE). 1 : [15][16] [15] f J[f] R[f] + λ f(θ) C (6) R[f] f (Fig. B ) R[f] = Γ(ϕ + ) Γ(ϕ ) = {Z(θ + ϕ + ) Z(θ + ϕ )}f(θ) (7) ϕ + ϕ Γ(ϕ) ϕ f(θ) C = 0 f 1 C C = 0 (6) λ (6) f(θ) C = 0 f R[f] f opt (f, λ) f (θ) = f opt [11] (4) (1)

4 4 74 (A) (C) (B) Fig. 3 f opt (θ) = 1 λ {Z(θ + ϕ +) Z(θ + ϕ )} (8) [15] λ = 1 Q/P P = f (θ), Q = {Z(θ+ ϕ + ) Z(θ + ϕ )} ϕ ± Z [15] ( ) R[f] S[f] = Γ (ϕ ) [16] 1 ( ) (8) ϕ ± (). : [17] 1 ϕ (4) 0 P1 ( f (θ) = P = ) 1:1 P ( ) 1:1 ( ) P3 1:1 m:n 1:1 P1 P P3 ( S1, S, S3 ) f p L p (S) f p f(θ) p 1 p = M <, (9) p 1 M p = (9) f = M f M p = 1 f = M f M

5 75 VOL. 0 NO. JUN p = (9) f = M f f(θ) (ess. sup) f(θ) < M ( θ S) f.1 (9) f(θ) = 0. (10),.1, maximize R[f] subject to f(θ) = 0, f p = M (11),.1, f p = M J[f] = R[f] + λ f(θ) f λ., J[f], (7) f g : J[f] = f(θ)[ Z(θ) + λ] f(θ)g(θ), (1) g(θ) = Z(θ)+λ, Z(θ) Z(θ + ϕ) Z(θ) ϕ ϕ + ϕ ϕ θ θ + ϕ θ. ( ) (11) R[f] S[f] = Γ (ϕ ) = f(θ)z (θ) (1) g(θ) g(θ) = Z (θ) g(θ) = Z(θ) + λ.1 J[f] (1) (Hölder s inequality)[18] fg 1 f p g q, (13) p q 1 p, q, p 1 +q 1 = 1 (13) (1) J[f] J[f] = fg fg = fg 1 f p g q = M g q (14) f g(= Z(θ) + λ) ( P1 P P3 )., Fig. C (a), (b), (c), 1 < p < ( p = ), p = 1, p = S1 1 < p < 1:1 ( f opt, p ) Lp(S) f opt, p Z (13) S p = 1 1:1 Z L 1 (S) ( f opt,1 ) S1 p 1 (13) p = 1, q = p = 1:1 ( f opt, ) L (S) Z S1 p p =, q = 1 S3 Z, f Z n, f m 1 Hodgkin-Huxley [16] Z Fig. C (d) Fig. 3C.

6 6 76 : Z(θ) = a 0 + j (a j cos jθ + b j sin jθ), f(θ) = c 0 + k (c k cos kθ + d k sin kθ), Z n (θ) a 0 + j (a nj cos njθ + b nj sin njθ), f m (θ) c 0 + k (c mk cos mkθ+d mk sin mkθ) 1 Γ m/n (ϕ) = Z(mθ + ϕ)f(nθ) Z, f Tsallis [19] [0] 100 Γ m/n (ϕ) = Z n (mθ + ϕ)f(nθ) = Z n (mθ + ϕ)f m (nθ), S1 S 1:1 Z Z n m:n f m (nθ) Fig. 3A Fig. 3B 1:1 1: 1: 1:1 Z n ( n = ), Fig. 3C Z n P1, P, P3 S1, S, S3 [17].1 [15][16] (1) () (3) m:n 3 [1][7][8][1][13] ( ) [1][] (p ),., ( [3]),,. Tsallis [4] 1 Z n, f m Z, f nj, mk m:1 (m:m=1:1 ) 1:1 B SCAT

7 77 VOL. 0 NO. JUN Kawasaki, K., Akiyama, Y., Komori, K., Uno, M., Takeuchi, H., Itagaki, T., Hino, Y., Kawasaki, Y., Ito, K. and Hajimiri, A., A millimeter-wave intra-connect solution, IEEE ISSCC Digest of Technical Paper, (010), ,,, 17 (007), Kuramoto, Y., Chemical Oscillations, Waves and Turbulence, Springer, Berlin, Adler, R., A Study of Locking Phenomena in Oscillators, Proc. IRE, 34[6] (1946), Kurokawa, K., Injection Locking of Microwave Solid-State Oscillators, Proc. IEEE, 61[10] (1973), Daikoku, K. and Mizushima, Y., Properties of Injection Locking in the Non-Linear Oscillator, Int. J. Electronics 31[3] (1971), Yamamoto, K. and Fujishima, M., A 44-µW 4.3- GHz Injection-Locked Frequency Divider with.3- GHz Locking Range, IEEE J. Solid-State Circuits, 40[3] (005), Kazimierczuk, M. K., Krizhanovski, V. G., Rassokhina, J. V. and Chernov, D. V., Injection-Locked Class-E Oscillator, IEEE Trans. Circuits and Systems I, 53[6] (006), Hoppenstea, F. C. and Izhikevich, E. M., Weakly Connected Neural Networks, Springer, New York, Hajimiri, A. and Lee, T. H., A General Theory of Phase Noise in Electrical Oscillators, IEEE J. Solid-State Circuits, 33[](1998), Bhansali, P. and Roychowdhury, J., Gen-Adler: the Generalized Adler s Equation for Injection Locking Analysis in Oscillators, in Proc. IEEE Asia and South Pacific Design Automation Conference, (009), Tanaka, H.-A., Hasegawa, A., Mizuno, H. and Endo, T., Synchronizability of distributed clock oscillators, IEEE Trans. Circuits and Systems I: Fundamental Theory and Applications, 49[9](00), Nagashima, T., Wei, X., Tanaka, H.-A. and Sekiya, H., Numerical derivations of locking ranges for injection-locked class-e oscillator, in Proc. IEEE 10th International Conference on Power Electronics and Drive Systems, (013), ,,,, A, 89[3](006), Harada, T., Tanaka, H.-A., Hankins, M. J. and Kiss, I. Z., Optimal Waveform for the Entrainment of a Weakly forced Ooscillator, Phys. Rev. Lett. 105[8](010), Zlotnik, A., Chen, Y., Kiss, I. Z., Tanaka, H.-A. and Li, J. S., Optimal Waveform for Fast Entrainment of Weakly Forced Nonlinear Oscillators, Phys. Rev. Lett., 111[](013), Tanaka, H.-A., Optimal entrainment with smooth, pulse, and square signals in weakly forced nonlinear oscillators, Physica D, vol. 88(014), Rudin, W., Real and Complex Analysis, third ed., McGraw-Hill, New York, 1987, Tsallis, C., Introduction to Nonextensive Statistical Mechanics, Springer, New York, Tanaka, H.-A., Synchronization limit of weakly forced nonlinear oscillators, J. Phys. A, 47[40], (014), ,, 17[](007), ,, :, 1[4](011), Li, J. S., Ruths, J., Yu, T. Y., Arthanari, H. and Wagner, G., Optimal pulse design in quantum control: A unified computational method, Proc. of the National Academy of Sciences, 108[5](011), Abstract In this article, a universal mechanism governing entrainment limit is shown to exist under weak forcings. This underlying mechanism enables us to understand how and why entrainability is maximized; maximization of the entrainment range or that of the stability of entrainment for general forcings including pulse trains, and a fundamental limit of general m:n entrainment, are clarified from a unified, global viewpoint. These entrainment limits are verified in the Hodgkin-Huxley neuron model as an example.

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